Second, we will establish the relation between the spaces appearing as a result of such extension and
the spaces \(\widetilde{B}^{l}_{p,q,r}(\mathbb {R}^{n},\lbrace t_{k}\rbrace )\) , which were introduced by the author in [1]} using the methods of nonlinear spline approximation.
The space \(\widetilde{B}^{l}_{p,p,r}(\mathbb {R}^{n},\lbrace \gamma _{k}\rbrace )\)  (with \(1\le r <p < \infty \) ) arises in the natural way as the trace space of the weighted Sobolev space
\(W^{l}_{p}(\mathbb {R}^{n}, \gamma )\)  with weight \(\gamma \in A^{\rm loc}_{\frac{p}{r}}(\mathbb {R}^{n})\)  (for details, see § 3 of [1]}).
Hence, the problem of finding an equivalent description of the space \(\widetilde{B}^{l}_{p,q,r}(\mathbb {R}^{n},\lbrace t_{k}\rbrace )\) 
is of great importance in some practical applications (for details see § 5).
